## How to single out the right atoms

For a quantum computer to be useful, its qubits have to be able to change their state in response to external stimuli. But when a large number of qubits are packed in a three-dimensional (3D) structure to optimize the use of space, altering one qubit can unintentionally change the state of others. Wang *et al.* devised a clever way to perform high-fidelity quantum gates only on intended qubits in a 3D array of Cs atoms. Although the operation initially changed the state of some of the other atoms, additional manipulation recovered their original state. The technique may be applicable to other quantum computing implementations.

*Science*, this issue p. 1562

## Abstract

Although the quality of individual quantum bits (qubits) and quantum gates has been steadily improving, the number of qubits in a single system has increased quite slowly. Here, we demonstrate arbitrary single-qubit gates based on targeted phase shifts, an approach that can be applied to atom, ion, or other atom-like systems. These gates are highly insensitive to addressing beam imperfections and have little cross-talk, allowing for a dramatic scaling up of qubit number. We have performed gates in series on 48 individually targeted sites in a 40% full 5 by 5 by 5 three-dimensional array created by an optical lattice. Using randomized benchmarking, we demonstrate an average gate fidelity of 0.9962(16), with an average cross-talk fidelity of 0.9979(2) (numbers in parentheses indicate the one standard deviation uncertainty in the final digits).

The performance of isolated quantum gates has recently been improved for several types of qubits, including trapped ions (*1*–*3*), Josephson junctions (*4*), quantum dots (*5*), and neutral atoms (*6*). Single-qubit gate errors now approach or, in the case of ions, surpass the commonly accepted error threshold (error per gate < 10^{–4}) (*7*, *8*) for fault-tolerant quantum computation (*9*–*12*). It remains a challenge in all these systems to execute targeted gates on many qubits in close physical proximity to one another with fidelities comparable with those for isolated qubits (*13*, *14*). Neutral atom and ion experiments have to date demonstrated the most qubits in the same system, 50 and 18 respectively (*15*, *17*). The highest-fidelity gates in these systems are based on microwave transitions, but addressing schemes typically depend on either addressing light beams (*6*, *15*, *18*–*20*), which are difficult to make as stable as microwaves, or magnetic field gradients (*2*, *21*), which limit the number of addressed qubits. Here, we present a way to induce phase shifts on atoms at targeted sites in a 5 by 5 by 5 optical lattice that is highly insensitive to addressing laser beam fluctuations. We further show how to convert targeted phase shifts into arbitrary single-qubit gates.

In previous work, we performed single-site addressing in a three-dimensional (3D) lattice using crossed laser beams, to selectively induce ac Stark shifts in target atoms, and microwaves, to temporarily map quantum states from a field-insensitive storage basis to the Stark-shifted computational basis (*15*). Although we used most of the same physical elements in this work, the crucial difference is that the gates described here are based on phase shifts in the storage basis and do not require transitions out of it. Nonresonant microwaves are applied that give opposite-sign ac Zeeman shifts (analogous to ac Stark shifts, but for magnetic dipole transitions) for different atoms. A specific series of nonresonant pulses and global π-pulses on the qubit transition gives a zero net phase shift for nontarget atoms and a controllable net phase shift for target atoms. The resultant gate fidelity is much better than our previous gate because of this gate’s extreme insensitivity to the addressing beam alignment and power, the insensitivity of the storage basis to magnetic fields and vector light shifts, and the independence on the phase of the nonresonant microwave pulses.

Detailed descriptions of our apparatus can be found in (*15*, *22*, *23*). We optically trapped and reliably imaged neutral ^{133}Cs atoms in a 5-μm spaced cubic optical lattice. The atoms were cooled to ~70% ground vibrational state occupancy and then microwave transferred into the qubit basis, the 6*S*_{1/2}, , and hyperfine sublevels, which we will call and , respectively. Lattice light spontaneous emission is the largest source of decoherence, with a 7 s coherence time () that is much longer than the typical microwave pulse time of 80 μs, which could be shortened with more microwave intensity. We detected the qubit states by clearing atoms in the states and imaging the atoms that remain.

To target an atom, we crossed at a right angle two circularly polarized, 880.250-nm addressing beams (beam waist, ~2.7 μm; Rayleigh range, ~26 μm). The addressing beams can be directed to a new target in <5 μs by using micro-electro-mechanical-system (MEMS) mirrors (*24*). The addressing beams induce only a modest ac Stark shift on the *m*_{F} = 0 qubit states (~400 Hz), but they cause a vector light shift on the *m*_{F} ≠ 0 levels. The vector light shifts are about twice as large for the target atom as for any other. This is illustrated in Fig. 1A, for which atoms are prepared in and a microwave near the transition is scanned. The resonances are visible for atoms at the intersection (orange, termed “cross” atoms), atoms in one addressing beam path (blue, termed “line” atoms), and the rest of the atoms (green, termed “spectator” atoms). The ac Stark shift for the line atoms, *f*, was chosen so that there is a region between the blue and orange peaks in which only a small fraction (2 × 10^{–4}) of atoms in any class (cross, line, or spectator) makes the transition. When a microwave pulse is applied in that frequency range, atoms experience different ac Zeeman shifts depending on their class.

The addressing pulse sequence for a pair of target atoms in two planes (Fig. 1C) consists of four stages (*15*). The qubit-resonant spin-echo pulses (Fig. 1C, black pulses on the microwave line) reverse the sign of the phase shifts, so that whatever phase shifts (ac Zeeman or ac Stark) a nontarget atom gets during the cross stages (Fig. 1C, stages 1 and 3) are exactly canceled by the shifts it gets during the dummy stages (Fig. 1C, stages 2 and 4), in which there is no cross atom. In contrast, the first target atom spends stage 1 as a cross, stage 3 as a spectator, and stages 2 and 4 as a line atom, and the second target atom spends stage 3 as a cross and stage 1 as a spectator. When the microwave frequency is chosen to be between the line and cross resonances, the change in the target atom’s status from cross to line changes the sign of the ac Zeeman shift. Away from the resonances, the net phase shift for the target atoms is (1)where δ is the microwave detuning from the spectator resonance, *k* is the shift of cross atoms in units of *f* (*16*), Ω (typically 4 kHz) is the microwave Rabi frequency, and *T* is the pulse duration. Successive terms correspond to the integrated ac Zeeman shift on the first target atom during successive stages. The overall phase shift can be directly controlled by changing the power of the microwave field. The black curve in Fig. 1B shows the result of an exact calculation of the target atom’s phase shift as a function of δ. The phase shift minimum at δ_{0} = 74.9 kHz is the preferred operating point for the gate because in the vicinity of δ_{0}, the shift depends quadratically on the change in δ and thus also on the addressing beams ac Stark shift, with the coefficient of 21 rad/(Δ*f*/*f*)^{2}. For example, a 2% change in *f* gives an 8-mrad phase shift, which in turn leads to only a 1 × 10^{–4} gate error. Because the intensity changes quadratically with beam alignment, the gate is sensitive to beam-pointing only at fourth order.

The phase shift on target sites amounts to a rotation about the *z* axis [an *R _{z}*(θ) gate], but a universal single-qubit gate requires arbitrary rotations about any arbitrary axis. We can make an

*R*(θ) gate by combining the

_{y}*R*(θ) gate with global rotations (2)For nontarget atoms, which see the global microwave pulses but experience no

_{z}*R*(θ), clearly has no net effect. It is straightforward to generalize this formula to obtain arbitrary rotations on a Bloch sphere for target atoms. The corresponding complete set of single-qubit gates on target atoms all leave the nontarget atoms unchanged.

_{z}We have demonstrated one *R _{z}*(π/2) gate on a sequence of 48 randomly chosen sites (in 24 gate pairs) within a 5 by 5 by 5 array. Given the average initial site occupancy of ~40%, an average of 20 qubits experience the phase gate during each implementation, whereas 30 remain in their original quantum superposition. We probed the coherence of all the atoms by closing the spin echo sequence with a global π/2 pulse whose phase we scanned (Fig. 2). In Fig. 2, the open diamonds indicate nontarget atoms, and the solid circles indicate atoms at the 48 target sites. The corresponding curves are sinusoidal fits to the data. The dashed lines mark the maximum and minimum populations one expects given perfect gate fidelity, , defined as the square of the projection of the measured state onto the intended state, in the face of state preparation and measurement (SPAM) errors (

*16*). From these curves, we determined that the error per gate pair, , is 25(13) × 10

^{–4}for both target and nontarget atoms on average (numbers in parentheses indicate the 1 SD uncertainty in the final digits). At least for this particular gate, most of the error is clearly common to target and nontarget atoms. The good contrast of the target atom fringe illustrates the homogeneity of the phase shifts across the ensemble. The error per gate, , is thus 13(7) × 10

^{–4}on average.

To graphically illustrate the flexibility of our gates, we have performed an *R _{z}*(π) rotation on a 32-site pattern (Fig. 3). We rotated the superpositions of a sequence of pairs of target atoms from to and then used a π/2 detection pulse, phase-shifted by π, to return spectator atoms to and target atoms to . Images from five planes summed over 50 implementations are shown in Fig. 3.

To confirm the robustness of this gate, we applied an *R _{z}*(π/2) gate to 24 targets and measured the probability that atoms return to after a π/2 pulse as a function of the fractional change in addressing beam intensity (Fig. 4, inset) (

*16*). The data confirm the theoretical prediction (solid line) of extreme insensitivity to addressing beam intensity (Fig. 1B).

To fully characterize gate performance, we used the standard randomized benchmarking (RB) protocol (*25*), which has been used in nuclear magnetic resonance (*26*), quantum dots (*5*), ion systems (*27*), and neutral atoms systems (*6*, *28*, *29*). We implemented RB by choosing computation gates (CG) at random from and Pauli gates (PG) at random from {*R _{x}*(±π),

*R*(±π),

_{y}*R*(±π),

_{z}*I*}, where

*I*is the identity operation (

*30*). After each randomized sequence, a detection gate is applied that in the absence of errors would return either the target qubits or the nontarget qubits to . For the targets, the average value of decays exponentially with the length of the randomized sequence (3)where

*d*/2 is the SPAM error and

_{if}*l*is the number of CG-PG operations applied to the pair of target atoms. A similar expression yields and for the spectator and line atoms, respectively—the unwanted changes wrought on the nontarget atoms when a random series of gates is executed on the targets. In this way, we can characterize cross-talk and extract errors per gate from errors per gate pair. For target atoms, ; for spectator atoms, ; and for line atoms, .

First analyzing the nontarget data (fig. S2), we determined that *d _{if}* = 0.11(1), , and . The cross-talk, defined as the average error per gate for nontarget atoms, is . Only global microwave pulse imperfections contribute to . Their contribution to may differ because the microwave sequence is optimized for spectators. Spontaneous emission from the addressing beams adds to , with a calculable contribution of 8 × 10

^{–4}. The RB data for the two target atoms are shown in Fig. 4. Using the

*d*from the nontargets, the fit of Eq. 3 to the data yields .

_{if}A summary of the errors for the target and nontarget atoms is shown in table S1. The errors can mostly be traced to microwave power stability (17 × 10^{–4} and 16 × 10^{–4}). We expect that reaching the state of the art (*2*, *27*), and perhaps tweaking our spin echo infrastructure (*31*) (fig. S1), can bring it below 10^{–4}. The next largest contribution is from the readily calculable spontaneous emission of addressing light (16 × 10^{–4} for target atoms). Its contribution is invariant with gate times because *f* must be changed inversely with gate time. Addressing with light between the D1 () and D2 () lines, as we do, locally minimizes spontaneous emission, but doubling the wavelength would further reduce spontaneous emission by a factor of 8 without dramatically compromising site-addressing. The required powerful addressing beams would exert a substantial force on line atoms, but deeper lattices and adiabatic addressing-beam turn-on would make the associated heating negligible. In our current experiment, we need to wait 70 μs after the addressing beams are turned on for our intensity lock to settle. Technical improvement there could halve the time the light is on, ultimately leaving the spontaneous emission contribution to the error just below 10^{–4} per gate. That limit is based on addressing with a differential light shift. It would disappear were the same basic scheme to be used on a three-level system, in which only one of the qubit states is strongly ac Stark–shifted by the addressing light. In that case, the phase-shifting microwaves (or light) would simply be off-resonant from the qubit transition, and the error caused by spontaneous emission would decrease proportional to the detuning. Because the dominant target errors have the same sources as the nontarget errors, improvements to the gate will correspondingly improve the cross-talk. Adapting this method to more common 1D and 2D geometries is straightforward. Only a single addressing beam is needed, and the dummy stages would be executed with half-intensity addressing beams. The same insensitivity to addressing beam power and alignment would follow.

Scalable addressing is an important step toward a scalable quantum computer. Future milestones that must be passed on the road to scalable neutral atom quantum computation include scalable addressing for two-qubit gates (*32*–*34*), reliable site filling (*19*), and the implementation of error correction (*7*).

## Supplementary Materials

## References and Notes

**Acknowledgments:**This work was supported by the U.S. National Science Foundation grant PHY-1520976. The data presented in this report are available on request to D.S.W.